He also spoke for the first time about the couple’s plans for the future, confirming that he would be leaving RAF Valley next month when “we have to move elsewhere.”

The Duke was visiting the Anglesey Show, a two-day agricultural event close to the rented farmhouse where he and the Duchess have lived for the past three years.

Speaking to Max and Maxine Davies, from Victor Harbour, near Adelaide, South Australia, he said: “George is doing well, thank you. We are all very hopeful of coming to Australia next year.”

Mr and Mrs Davies, aged 77 and 75 respectively, said they were thrilled at the news.

“We are on holiday here and can’t believe we got to talk to him,” said Mrs Davies. “How wonderful that the family will come to Australia to visit.”

The Prince and Princess of Wales broke with tradition in 1983 by taking the baby Prince William with them to Australia on their first royal tour after their marriage. Although there has been widespread speculation that the Duke and Duchess might follow suit, it is the first time the Duke has admitted plans are under way.

Caroline Roberts, from Anglesey, also asked the Duke about the baby, and said: “He said he was fine but had his moments.”

In a speech to the crowd, which he began in Welsh, he said: “Thank you, people of Anglesey. It is a great pleasure to be here. I am so proud to have lived on Anglesey, the Mother of Wales. I have even learnt a little Welsh.

“My Welsh pronunciation is bad but it is slowly getting better, I hope.

“Catherine is sorry that she cannot be here today at the County Show, but she and George would have loved to have been here. He’s pretty loud but, of course, very good looking.

“I have to say that I thought search and rescue duties over Snowdonia were physically and mentally demanding, but looking after a three-week-old baby is up there.

“I know that I speak for Catherine when I say that I have never in my life known somewhere as beautiful and as welcoming as Anglesey.

“The views across the Menai Straits are undoubtedly among the most stunning in the British Isles. I know that both of us will miss it terribly when my search and rescue tour of duty comes to an end next month and we have to move elsewhere.

“This island has been our first home together, and it will always be an immensely special place for us both. Catherine and I look forward to returning again and again over the coming years with our family.”

A royal spokesman said the Duke had not been having formal lessons in Welsh, but his Welsh-speaking friends helped him. His reference to Anglesey as the Mother of Wales came from the Middle Ages, when it was thought that the island’s fields could feed the nation.

As he toured the show the Duke spoke to Joan Roberts, 70, from Carmel, North Wales, about his new duties as a father.

She said: “He told me that he hopes [George] will sleep through the night soon. It was the first time I have met him. It was really lovely.”

Later, the Duke judged a cattle show. An announcement on his future career will be made in the next month, before his tour of duty at RAF Valley finishes in the middle of September.